37:
1209:. If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing the number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected. Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value.
1065:
731:
724:
689:
717:
710:
703:
696:
682:
675:
668:
661:
655:
508:, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic.
600:
one number (one) which is neither of the two. Similarly, in form, the right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.
599:
It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers
1087:
into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable
1124:
describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its
101:
is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the
86:
The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings.
877:: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other.
1111:
is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.
1236:, the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See
106:
is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even.
490:
425:
355:
307:
455:
387:
258:
1137:
of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number.
220:
180:
141:
968:
938:
1356:
1096:, and other twisting puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the
595:
instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches the philosophical afterthought,
555:
The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even
1247:
are chosen so that the houses on one side of a street have even numbers and the houses on the other side have odd numbers. Similarly, among
1161:
is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game
1255:, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights.
90:
Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the
1743:
311:
1251:, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways. Among airline
269:
1869:
1844:
1635:
1535:
884:
may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.
847:
of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the
1125:
result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case
1530:, Grundlehren der Mathematischen Wissenschaften , vol. 290 (3rd ed.), New York: Springer-Verlag, p. 10,
2130:
The Big Roads: The Untold Story of the
Engineers, Visionaries, and Trailblazers Who Created the American Superhighways
2165:
2138:
2111:
2084:
2057:
2030:
1931:
1904:
1819:
1792:
1726:
1692:
1665:
1573:
1488:
1461:
1427:
1400:
1339:
1312:
1097:
1248:
1052:
calculations have shown this conjecture to be true for integers up to at least 4 × 10, but still no general
83:
by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 7, and 21 are odd numbers.
460:
395:
225:
2191:
1371:
1027:
element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0
2186:
1778:
191:
150:
874:
861:, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in
117:
1864:, London Mathematical Society Lecture Note Series, vol. 272, Cambridge: Cambridge University Press,
1839:, London Mathematical Society Lecture Note Series, vol. 188, Cambridge: Cambridge University Press,
1084:
492:
where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below.
1744:"Empirical verification of the even Goldbach conjecture, and computation of prime gaps, up to 4·10"
1787:, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, pp. 26–27,
1682:
1048:
states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern
1031:
this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.
94:
is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the
1146:
1104:
1045:
438:
370:
1237:
1174:
1076:
587:, to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century:
428:
17:
1954:, Math. Sci. Res. Inst. Publ., vol. 29, Cambridge: Cambridge Univ. Press, pp. 61–78,
1121:
1024:
988:
568:
64:
2101:
2074:
1921:
1894:
1809:
1712:
1655:
1563:
1478:
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1390:
1329:
2155:
2128:
2020:
1964:
1782:
1625:
1525:
1451:
1302:
1225:
1154:
881:
572:
103:
2047:
1511:
1959:
1879:
1854:
1545:
1206:
848:
1978:
543:
By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the
47:
be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green)
8:
1716:
947:
917:
908:
900:
588:
2001:
1811:
Adventures in Group Theory: Rubik's Cube, Merlin's
Machine, and Other Mathematical Toys
1606:
1198:
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1068:
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1028:
584:
505:
2161:
2134:
2107:
2080:
2053:
2026:
1927:
1900:
1865:
1840:
1815:
1788:
1722:
1688:
1661:
1631:
1569:
1531:
1484:
1457:
1423:
1396:
1335:
1308:
432:
263:
24:
1763:
1565:
Chess
Thinking: The Visual Dictionary of Chess Moves, Rules, Strategies and Concepts
36:
1993:
1758:
1708:
1598:
1559:
1505:
1080:
1023:
of integers, but the odd numbers do not—this is clear from the fact that the
892:
873:
alternate parity between moves. This form of parity was famously used to solve the
1955:
1875:
1850:
1541:
1244:
1213:
1166:
1089:
858:
844:
40:
1443:
1947:
1651:
1233:
1039:
1020:
870:
866:
835:
831:
564:
98:
91:
1510:, translated by Jarvis, Josephine, New York: A Lovell & Company, pp.
2180:
1997:
1447:
1331:
A Walk
Through Combinatorics: An Introduction to Enumeration and Graph Theory
1252:
1134:
559:
odd, since the concepts of even and odd apply only to integers. But when the
390:
1269:
1108:
1035:
501:
1173:, so its value is zero for evil numbers and one for odious numbers. The
996:
364:
56:
2005:
1064:
609:
1610:
1229:
1202:
1016:
1228:. (With cylindrical pipes open at both ends, used for example in some
869:
are constrained to moving between squares of the same parity, whereas
1216:
with a cylindrical bore and in effect closed at one end, such as the
185:
80:
1624:
Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997),
1602:
1221:
1217:
1093:
1049:
560:
1742:
Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2013),
1264:
1169:
maps a number to the number of 1's in its binary representation,
95:
68:
1681:
Lial, Margaret L.; Salzman, Stanley A.; Hestwood, Diana (2005),
1162:
1042:
are even; it is unknown whether any odd perfect numbers exist.
1721:, MAA Spectrum, Cambridge University Press, pp. 242–244,
23:"Odd number" redirects here. For the 1962 Argentine film, see
2160:, Corporations that changed the world, ABC-CLIO, p. 90,
912:
862:
544:
1741:
1177:, an infinite sequence of 0's and 1's, has a 0 in position
1038:
are odd, with one exception: the prime number 2. All known
500:
The following laws can be verified using the properties of
1205:
appended to a binary number provides the simplest form of
865:, where the parity of a square is indicated by its color:
851:
and its higher-dimensional that is generalizations, the
1623:
147:
is an integer; an odd number is an integer of the form
1003:
is even or odd if and only if its numerator is so in
950:
920:
834:
is confined to squares of the same parity; the black
610:
Higher dimensions and more general classes of numbers
463:
441:
398:
373:
314:
272:
266:
of even and odd numbers can be defined as following:
228:
194:
153:
120:
1680:
184:An equivalent definition is that an even number is
2022:A Student's Guide to Coding and Information Theory
1808:Joyner, David (2008), "13.1.2 Parity conditions",
1589:Mendelsohn, N. S. (2004), "Tiling with dominoes",
1153:is a number that has an even number of 1's in its
962:
932:
484:
449:
419:
381:
349:
301:
252:
214:
174:
135:
2099:
1893:Gustafson, Roy David; Hughes, Jeffrey D. (2012),
1364:The Pentagon: A Mathematics Magazine for Students
2178:
1899:(11th ed.), Cengage Learning, p. 315,
1834:
1442:
838:can only jump to squares of alternating parity.
615:
2100:Cromley, Ellen K.; McLafferty, Sara L. (2011),
1892:
2025:, Cambridge University Press, pp. 19–20,
1133:) = 0, to be both odd and even. The
2106:(2nd ed.), Guilford Press, p. 100,
1687:(7th ed.), Addison Wesley, p. 128,
1523:
1140:
344:
315:
296:
273:
1919:
1835:Bender, Helmut; Glauberman, George (1994),
1480:Ancient Greek Philosophy: Thales to Gorgias
1307:, Pearson Education India, pp. 20–21,
1300:
511:
431:. Parity can then be defined as the unique
1862:Character theory for the odd order theorem
1859:
1588:
1558:
1415:
1392:Mathematics for Elementary School Teachers
1192:
485:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
420:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
350:{\displaystyle \{2k+1:k\in \mathbb {Z} \}}
2133:, Houghton Mifflin Harcourt, p. 95,
2018:
1976:
1926:, Alpha Science Int'l Ltd., p. 853,
1762:
1650:
1630:, ClassicalRealAnalysis.com, p. 37,
1388:
478:
465:
443:
413:
400:
375:
340:
292:
114:An even number is an integer of the form
51:be evenly divided in 2 by 3 (lime green).
2019:Moser, Stefan M.; Chen, Po-Ning (2012),
1979:"Evil twins alternate with odious twins"
1837:Local analysis for the odd order theorem
1568:, Simon and Schuster, pp. 273–274,
1524:Conway, J. H.; Sloane, N. J. A. (1999),
1483:, Pearson Education India, p. 126,
1419:The A to Z of Mathematics: A Basic Guide
1063:
302:{\displaystyle \{2k:k\in \mathbb {Z} \}}
35:
2072:
1952:Games of no chance (Berkeley, CA, 1994)
1920:Jain, R. K.; Iyengar, S. R. K. (2007),
1777:
1503:
1185:is evil, and a 1 in that position when
730:
723:
688:
33:Property of being an even or odd number
2179:
2045:
1807:
1707:
1476:
1422:, John Wiley & Sons, p. 181,
716:
709:
702:
695:
681:
674:
667:
660:
504:. They are a special case of rules in
2153:
2126:
1296:
1294:
1292:
1290:
1288:
1286:
1284:
651:
604:
583:The ancient Greeks considered 1, the
1527:Sphere packings, lattices and groups
1354:
1327:
1946:
1453:Notes on Introductory Combinatorics
13:
1281:
1224:produced are odd multiples of the
14:
2203:
1395:, Cengage Learning, p. 198,
1334:, World Scientific, p. 178,
1083:) is the parity of the number of
843:Integer coordinates of points in
527:
1923:Advanced Engineering Mathematics
1010:
729:
722:
715:
708:
701:
694:
687:
680:
673:
666:
659:
653:
2147:
2120:
2093:
2066:
2039:
2012:
1970:
1940:
1913:
1886:
1828:
1814:, JHU Press, pp. 252–253,
1801:
1771:
1764:10.1090/s0025-5718-2013-02787-1
1735:
1701:
1674:
1644:
1617:
1591:The College Mathematics Journal
1582:
1552:
1301:Vijaya, A.V.; Rodriguez, Dora,
1249:United States numbered highways
1059:
589:Friedrich Wilhelm August Fröbel
563:is an integer, it will be even
1517:
1497:
1470:
1436:
1416:Sidebotham, Thomas H. (2003),
1409:
1382:
1348:
1321:
944:, while elements of the coset
240:
202:
79:. An integer is even if it is
1:
1275:
495:
109:
2076:An Introduction to Acoustics
1456:, Springer, pp. 21–22,
875:mutilated chessboard problem
450:{\displaystyle \mathbb {Z} }
382:{\displaystyle \mathbb {Z} }
233:
7:
2073:Randall, Robert H. (2005),
1950:(1996), "Impartial games",
1860:Peterfalvi, Thomas (2000),
1504:Froebel, Friedrich (1885),
1450:; Woods, Donald R. (2009),
1258:
1115:
1105:Feit–Thompson theorem
882:parity of an ordinal number
849:face-centered cubic lattice
550:
10:
2208:
1751:Mathematics of Computation
1370:(2): 17–20, archived from
578:
253:{\displaystyle 2\not |\ x}
222:and an odd number is not:
22:
15:
1977:Bernhardt, Chris (2009),
1684:Basic College Mathematics
1660:, Springer, p. 199,
1657:Elements of Number Theory
1147:combinatorial game theory
1141:Combinatorial game theory
1015:The even numbers form an
1998:10.4169/193009809x469084
1304:Figuring Out Mathematics
999:(2). Then an element of
512:Addition and subtraction
2052:, Springer, p. 4,
2046:Berrou, Claude (2011),
1389:Bassarear, Tom (2010),
1357:"Divisibility in bases"
1238:harmonic series (music)
1220:at the mouthpiece, the
1193:Additional applications
1077:parity of a permutation
545:field with two elements
429:field with two elements
215:{\displaystyle 2\ |\ x}
175:{\displaystyle x=2k+1.}
18:Parity (disambiguation)
2079:, Dover, p. 181,
1355:Owen, Ruth L. (1992),
1072:
964:
934:
911:is 2. Elements of the
602:
486:
451:
421:
383:
351:
303:
254:
216:
176:
137:
52:
2192:Elementary arithmetic
2154:Lauer, Chris (2010),
2103:GIS and Public Health
2049:Codes and turbo codes
1328:Bóna, Miklós (2011),
1226:fundamental frequency
1155:binary representation
1067:
1046:Goldbach's conjecture
974:. As an example, let
965:
935:
597:
487:
452:
422:
384:
352:
304:
255:
217:
177:
138:
104:binary numeral system
39:
2187:Parity (mathematics)
2127:Swift, Earl (2011),
1986:Mathematics Magazine
1963:. See in particular
1507:The Education of Man
1207:error detecting code
1122:parity of a function
948:
918:
593:The Education of Man
461:
439:
396:
371:
312:
270:
226:
192:
151:
136:{\displaystyle x=2k}
118:
16:For other uses, see
1718:Mathematical Cranks
1243:In some countries,
1175:Thue–Morse sequence
1098:configuration space
963:{\displaystyle 1+I}
933:{\displaystyle 0+I}
533:even × even = even;
517:even ± even = even;
237:
2157:Southwest Airlines
1784:Permutation Groups
1757:(288): 2033–2060,
1199:information theory
1100:of these puzzles.
1073:
960:
930:
830:Each of the white
605:Higher mathematics
575:than the divisor.
536:even × odd = even;
506:modular arithmetic
482:
447:
417:
379:
347:
299:
250:
212:
172:
133:
53:
1871:978-0-521-64660-4
1846:978-0-521-45716-3
1779:Cameron, Peter J.
1713:"Perfect numbers"
1709:Dudley, Underwood
1637:978-0-13-458886-5
1560:Pandolfini, Bruce
1537:978-0-387-98585-5
1448:Tarjan, Robert E.
828:
827:
523:odd ± odd = even;
520:even ± odd = odd;
433:ring homomorphism
246:
236:
208:
200:
71:of whether it is
2199:
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1214:wind instruments
1081:abstract algebra
1056:has been found.
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845:Euclidean spaces
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539:odd × odd = odd;
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1981:
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1948:Guy, Richard K.
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1896:College Algebra
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1603:10.2307/4146865
1587:
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1477:Tankha (2006),
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1167:parity function
1143:
1118:
1088:definition. In
1079:(as defined in
1071:in solved state
1069:Rubik's Revenge
1062:
1040:perfect numbers
1013:
985:
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949:
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223:
201:
193:
190:
189:
152:
149:
148:
119:
116:
115:
112:
41:Cuisenaire rods
34:
31:
21:
12:
11:
5:
2205:
2195:
2194:
2189:
2174:
2173:
2166:
2146:
2139:
2119:
2112:
2092:
2085:
2065:
2058:
2038:
2031:
2011:
1969:
1939:
1932:
1912:
1905:
1885:
1870:
1845:
1827:
1820:
1800:
1793:
1770:
1734:
1727:
1700:
1693:
1673:
1666:
1643:
1636:
1616:
1597:(2): 115–120,
1581:
1574:
1551:
1536:
1516:
1496:
1489:
1469:
1462:
1435:
1428:
1408:
1401:
1381:
1347:
1340:
1320:
1313:
1279:
1277:
1274:
1273:
1272:
1267:
1260:
1257:
1253:flight numbers
1194:
1191:
1142:
1139:
1117:
1114:
1107:states that a
1085:transpositions
1061:
1058:
1012:
1009:
983:
970:may be called
959:
956:
953:
940:may be called
929:
926:
923:
854:
829:
826:
825:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
795:
792:
788:
787:
784:
780:
779:
776:
772:
771:
768:
764:
763:
760:
756:
755:
752:
748:
747:
744:
740:
739:
736:
728:
721:
714:
707:
700:
693:
686:
679:
672:
665:
658:
652:
650:
646:
645:
643:
640:
637:
634:
631:
628:
625:
622:
619:
614:
613:
611:
608:
606:
603:
580:
577:
573:factors of two
565:if and only if
552:
549:
541:
540:
537:
534:
529:
528:Multiplication
526:
525:
524:
521:
518:
513:
510:
497:
494:
480:
476:
472:
467:
445:
415:
411:
407:
402:
377:
346:
342:
338:
335:
332:
329:
326:
323:
320:
317:
298:
294:
290:
287:
284:
281:
278:
275:
249:
242:
231:
211:
204:
197:
171:
168:
165:
162:
159:
156:
132:
129:
126:
123:
111:
108:
99:numeral system
92:parity of zero
32:
9:
6:
4:
3:
2:
2204:
2193:
2190:
2188:
2185:
2184:
2182:
2169:
2167:9780313378638
2163:
2159:
2158:
2150:
2142:
2140:9780547549132
2136:
2132:
2131:
2123:
2115:
2113:9781462500628
2109:
2105:
2104:
2096:
2088:
2086:9780486442518
2082:
2078:
2077:
2069:
2061:
2059:9782817800394
2055:
2051:
2050:
2042:
2034:
2032:9781107015838
2028:
2024:
2023:
2015:
2007:
2003:
1999:
1995:
1991:
1987:
1980:
1973:
1966:
1961:
1957:
1953:
1949:
1943:
1935:
1933:9781842651858
1929:
1925:
1924:
1916:
1908:
1906:9781111990909
1902:
1898:
1897:
1889:
1881:
1877:
1873:
1867:
1863:
1856:
1852:
1848:
1842:
1838:
1831:
1823:
1821:9780801897269
1817:
1813:
1812:
1804:
1796:
1794:9780521653787
1790:
1786:
1785:
1780:
1774:
1765:
1760:
1756:
1752:
1745:
1738:
1730:
1728:9780883855072
1724:
1720:
1719:
1714:
1710:
1704:
1696:
1694:9780321257802
1690:
1686:
1685:
1677:
1669:
1667:9780387955872
1663:
1659:
1658:
1653:
1647:
1639:
1633:
1629:
1628:
1627:Real Analysis
1620:
1612:
1608:
1604:
1600:
1596:
1592:
1585:
1577:
1575:9780671795023
1571:
1567:
1566:
1561:
1555:
1547:
1543:
1539:
1533:
1529:
1528:
1520:
1513:
1509:
1508:
1500:
1492:
1490:9788177589399
1486:
1482:
1481:
1473:
1465:
1463:9780817649524
1459:
1455:
1454:
1449:
1445:
1444:Pólya, George
1439:
1431:
1429:9780471461630
1425:
1421:
1420:
1412:
1404:
1402:9780840054630
1398:
1394:
1393:
1385:
1377:on 2015-03-17
1373:
1369:
1365:
1358:
1351:
1343:
1341:9789814335232
1337:
1333:
1332:
1324:
1316:
1314:9788131703571
1310:
1306:
1305:
1297:
1295:
1293:
1291:
1289:
1287:
1285:
1280:
1271:
1268:
1266:
1263:
1262:
1256:
1254:
1250:
1246:
1241:
1239:
1235:
1234:open diapason
1231:
1227:
1223:
1219:
1215:
1210:
1208:
1204:
1200:
1190:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1159:odious number
1156:
1152:
1148:
1138:
1136:
1135:Taylor series
1132:
1128:
1123:
1113:
1110:
1106:
1101:
1099:
1095:
1091:
1086:
1082:
1078:
1070:
1066:
1057:
1055:
1051:
1047:
1043:
1041:
1037:
1036:prime numbers
1032:
1030:
1026:
1022:
1018:
1011:Number theory
1008:
1006:
1002:
998:
994:
990:
982:
978:
973:
957:
954:
951:
943:
927:
924:
921:
914:
910:
906:
902:
898:
894:
890:
885:
883:
878:
876:
872:
868:
864:
860:
857:
850:
846:
837:
833:
824:
821:
818:
815:
812:
809:
806:
803:
800:
798:
797:
793:
790:
789:
785:
782:
781:
777:
774:
773:
769:
766:
765:
761:
758:
757:
753:
750:
749:
745:
742:
741:
737:
648:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
618:
617:
601:
596:
594:
590:
586:
576:
574:
570:
566:
562:
558:
548:
546:
538:
535:
532:
531:
522:
519:
516:
515:
509:
507:
503:
493:
474:
470:
434:
430:
409:
405:
392:
391:quotient ring
366:
363:numbers is a
362:
357:
336:
333:
330:
327:
324:
321:
318:
288:
285:
282:
279:
276:
265:
260:
247:
229:
209:
195:
187:
182:
169:
166:
163:
160:
157:
154:
146:
130:
127:
124:
121:
107:
105:
100:
97:
93:
88:
84:
82:
78:
74:
70:
66:
62:
58:
50:
46:
43:: 5 (yellow)
42:
38:
29:
27:
19:
2156:
2149:
2129:
2122:
2102:
2095:
2075:
2068:
2048:
2041:
2021:
2014:
1992:(1): 57–62,
1989:
1985:
1972:
1951:
1942:
1922:
1915:
1895:
1888:
1861:
1836:
1830:
1810:
1803:
1783:
1773:
1754:
1750:
1737:
1717:
1703:
1683:
1676:
1656:
1646:
1626:
1619:
1594:
1590:
1584:
1564:
1554:
1526:
1519:
1506:
1499:
1479:
1472:
1452:
1438:
1418:
1411:
1391:
1384:
1372:the original
1367:
1363:
1350:
1330:
1323:
1303:
1270:Half-integer
1242:
1232:such as the
1211:
1196:
1186:
1182:
1178:
1158:
1150:
1144:
1130:
1126:
1119:
1109:finite group
1102:
1090:Rubik's Cube
1074:
1060:Group theory
1044:
1033:
1014:
1004:
1000:
992:
989:localization
980:
976:
971:
941:
904:
896:
888:
886:
879:
852:
842:
598:
592:
582:
556:
554:
542:
502:divisibility
499:
360:
358:
261:
183:
144:
113:
89:
85:
76:
72:
60:
54:
48:
44:
25:
1768:. In press.
1230:organ stops
1189:is odious.
1151:evil number
997:prime ideal
365:prime ideal
359:The set of
57:mathematics
2181:Categories
1965:p. 68
1276:References
1203:parity bit
496:Properties
110:Definition
26:Odd Number
1222:harmonics
1157:, and an
571:has more
337:∈
289:∈
186:divisible
81:divisible
2006:27643161
1781:(1999),
1711:(1992),
1654:(2003),
1562:(1995),
1259:See also
1218:clarinet
1171:modulo 2
1116:Analysis
1094:Megaminx
1050:computer
1025:identity
895:and let
859:lattices
591:'s 1826
569:dividend
561:quotient
551:Division
389:and the
235:⧸
65:property
1960:1427957
1880:1747393
1855:1311244
1611:4146865
1546:1662447
1265:Divisor
1019:in the
995:at the
987:be the
871:knights
867:bishops
832:bishops
579:History
427:is the
96:decimal
69:integer
63:is the
2164:
2137:
2110:
2083:
2056:
2029:
2004:
1958:
1930:
1903:
1878:
1868:
1853:
1843:
1818:
1791:
1725:
1691:
1664:
1634:
1609:
1572:
1544:
1534:
1487:
1460:
1426:
1399:
1338:
1311:
1165:. The
1163:Kayles
1029:modulo
907:whose
899:be an
836:knight
245:
207:
199:
188:by 2:
143:where
67:of an
61:parity
45:cannot
28:(film)
2002:JSTOR
1982:(PDF)
1747:(PDF)
1607:JSTOR
1375:(PDF)
1360:(PDF)
1181:when
1149:, an
1054:proof
1017:ideal
913:coset
909:index
901:ideal
891:be a
863:chess
585:monad
435:from
2162:ISBN
2135:ISBN
2108:ISBN
2081:ISBN
2054:ISBN
2027:ISBN
1928:ISBN
1901:ISBN
1866:ISBN
1841:ISBN
1816:ISBN
1789:ISBN
1723:ISBN
1689:ISBN
1662:ISBN
1632:ISBN
1570:ISBN
1532:ISBN
1485:ISBN
1458:ISBN
1424:ISBN
1397:ISBN
1336:ISBN
1309:ISBN
1201:, a
1120:The
1103:The
1075:The
1034:All
1021:ring
942:even
887:Let
880:The
567:the
361:even
264:sets
262:The
73:even
1994:doi
1759:doi
1599:doi
1512:240
1212:In
1197:In
1145:In
991:of
984:(2)
972:odd
903:of
557:nor
457:to
367:of
77:odd
75:or
55:In
49:can
2183::
2000:,
1990:82
1988:,
1984:,
1956:MR
1876:MR
1874:,
1858:;
1851:MR
1849:,
1755:83
1753:,
1749:,
1715:,
1605:,
1595:35
1593:,
1542:MR
1540:,
1446:;
1368:51
1366:,
1362:,
1283:^
1240:.
1092:,
1007:.
979:=
547:.
170:1.
59:,
2171:.
2144:.
2117:.
2090:.
2063:.
2036:.
2009:.
1996::
1967:.
1937:.
1910:.
1883:.
1825:.
1798:.
1761::
1732:.
1698:.
1671:.
1641:.
1614:.
1601::
1579:.
1549:.
1494:.
1467:.
1433:.
1406:.
1379:.
1345:.
1318:.
1187:i
1183:i
1179:i
1131:x
1129:(
1127:f
1005:Z
1001:R
993:Z
981:Z
977:R
958:I
955:+
952:1
928:I
925:+
922:0
905:R
897:I
889:R
855:n
853:D
822:h
819:g
816:f
813:e
810:d
807:c
804:b
801:a
794:1
791:1
786:2
783:2
778:3
775:3
770:4
767:4
762:5
759:5
754:6
751:6
746:7
743:7
738:8
649:8
642:h
639:g
636:f
633:e
630:d
627:c
624:b
621:a
479:Z
475:2
471:/
466:Z
444:Z
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410:2
406:/
401:Z
376:Z
345:}
341:Z
334:k
331::
328:1
325:+
322:k
319:2
316:{
297:}
293:Z
286:k
283::
280:k
277:2
274:{
248:x
241:|
230:2
210:x
203:|
196:2
167:+
164:k
161:2
158:=
155:x
145:k
131:k
128:2
125:=
122:x
30:.
20:.
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